1. 1 is a natural number.
2. Every natural number is equal to itself (equality is reflexive).
3. For all natural numbers a and b, a = b if and only if b = a (equality is symmetric).
4. For all natural numbers a, b, and c, if a = b and b = c then a = c (equality is transitive).
5. If a = b and b is a natural number then a is a natural number.
6. If a is a natural number then Sa is a natural number.
7. If a and b are natural numbers then a = b if and only if Sa = Sb.
8. If a is a natural number then Sa is not equal to 1.
9. For every set K, if 1 is in K and for every natural number x in K, Sx is also in K, then every natural number is in K. (It makes no difference here whether all elements of K are natural numbers.)